Projective Geometric Algebra

Projective geometric algebra is a four-dimensional mathematical model that naturally incorporates representations for Euclidean points, lines, and planes as well as operations for performing rotations, reflections, and translations in a single algebraic structure that is a specific type of Clifford algebra. It completely subsumes conventional models that include homogeneous coordinates, Plücker coordinates, quaternions, and screw theory (which makes use of dual quaternions). This makes projective geometric algebra a natural fit for areas of computer science that routinely use these mathematical concepts, especially computer graphics and robotics.

This page is a central resource containing all of the work on the subject of geometric algebra by Dr. Eric Lengyel. Some of the most recent developments in projective geometric algebra can be found in his blog:

Projective Geometric Algebra Done Right

Symmetries in Projective Geometric Algebra

Much more information is available on the Projective Geometric Algebra Wiki.

A C++ library that implements much of this math is available under a GPL license on GitHub.

The 36×24 inch reference poster below contains a huge amount of information, including new research from 2020–2021.

• A high-quality print is available on Amazon.

• There is also a PDF version.

Projective Geometric Algebra
27D1 27C7

This website is an official silver sponsor of the symbols ⟑ and ⟇, which have Unicode values U+27D1 and U+27C7. These symbols denote the geometric product and geometric antiproduct in projective geometric algebra.

Foundations of Game Engine Development, Volume 1: Mathematics

This book, written in 2016, contains an entire chapter about projective geometry in Grassmann algebra. It is a detailed introduction to the subject that is the perfect starting place for anyone who wants to learn details about the wedge and antiwedge products.

Foundations of Game Engine Development website

Foundations of Game Engine Development, Volume 1: Mathematics

New Developments in Projective Geometric Algebra

These are the slides from the talk about PGA at the Game Developers Conference 2021.

Download slides as PDF

New Developments in Projective Geometric Algebra
Grassmann Algebra in Game Development

Grassmann Algebra in Game Development

These are the slides from the talks about Grassmann algebra given at the Game Developers Conference in 2012 and 2014.

Download slides as PDF

A Bigger Mathematical Picture for Computer Graphics

These are the slides from the keynote talk about Grassmann algebra at the WSCG conference in 2012.

Download slides as PDF

Slides and audio recording on YouTube

A Bigger Mathematical Picture for Computer Graphics